The optical micro-lithography process in semiconductor fabrication, also known as the photolithography process, involves the reproduction of desired circuit patterns onto semiconductor wafers for an overall desired circuit performance. The desired circuit patterns are typically represented as apertures with dimensionally compensated shapes formed on a template commonly referred to as a photomask, where the dimensional compensation aims to provide the desired circuit features on the wafer. In optical micro-lithography, patterns on the photo-mask template are projected onto a photo-resist coated wafer by way of optical imaging through an exposure system.
The continuous advancement of VLSI chip manufacturing technology to meet Moore's law of shrinking device dimensions in geometric progression has spurred the development of Resolution Enhancement Techniques (RET), Optical Proximity Correction (OPC) methodologies, Inverse Lithography Technology (ILT), and Source Mask Optimization (SMO) in optical microlithography. These techniques aim to provide mask patterns that are dimensionally compensated to correct for the errors that arise when forming images of mask shapes which are barely resolvable by the projection optical system. The limited resolution causes the wafer locations where a feature edge is desired in the developed photoresist to actually be exposed by “spillover” light from the images of adjacent features, and the detailed shape of the resulting exposed image must be determined in order to provide proper dimensional compensation in the mask aperture shapes. The printed wafer shapes are also influenced by non-ideal development behavior in the photoresist, but this too is determined by the detailed image pattern within the neighborhood of a given feature. The images projected by the optical system are of the partially coherent kind, meaning that the illumination source pattern consists of many independent illuminating waves rather than a purely coherent beam, i.e. the source distribution has a complicated shape in directional space, with the illuminating waves not being so complete in their directional coverage as to flood-illuminate the mask, which would produce incoherent images. While the directional distribution has a complex shape which is chosen by methods well known in the art, the total intensity field produced by the illuminating waves as they overlap on the mask is generally made highly uniform; thus it is the partial coherence distribution of the illuminating beams rather than their intensity distribution which is designed to enhance resolution.
The RET techniques based on partially coherent illumination are expected to be used by chip manufacturers for the foreseeable future due to the high volume yield in manufacturing and extended resolution that they provide, and their general past history of success. However, the ever shrinking device dimensions combined with the desire to enhance circuit performance in the deep sub-wavelength domain require ever more compute intensive applications of OPC and related methodologies to ensure the fidelity of mask patterns on the printed wafer as device counts increase. Methods to provide these capabilities are generally referred to as computational lithography. Device counts in individual integrated circuit levels now often exceed one billion, and providing dimensionally compensated patterns on this scale is quite expensive using known methods. In recent decades a substantial commercial industry has developed to implement and apply these methods in an efficient manner.
For the most part all of these methods use the same class of physical model to define the impact of resolution loss in projecting the mask shapes, namely the Hopkins model that is known to govern the complex and nonlinear partially coherent imaging process. In addition, an approximate but computationally more efficient form of the standard Hopkins model is universally applied in order to approximately match the partially coherent imaging process when mask shapes must be provided at full chip scale, namely the Optimal Coherent Systems approximation, which, as will be described, approximates the complex partially coherent imaging process by forming a superposition of simpler coherent images that can be determined far more quickly. The underlying physical basis for these methods is expressed in the well-known Hopkins equation of partially coherent imaging, which determines the intensity at a given image point from a sum of contributions from all pairs of points in the vicinity of the conjugate mask point, or from all pairs of spatial frequencies that diffract from within that region of the mask. Because of this pairwise interaction, partially coherent imaging takes place within a doubled domain (and correspondingly the Hopkins equation operates over a doubled domain), i.e., partially coherent images are in effect projected from a doubling of the space in which the mask patterns are defined, and in the Hopkins model the image intensity thus has a quadratic dependence on the pattern content (more specifically a bilinear dependence). The bilinear kernel that expresses the image contribution from a pair of interfering points or spatial frequencies is known as the transmission cross coefficient (tcc for interfering pairs of points, or TCC for interfering frequencies). Because of its quadratic nonlinearity the Hopkins equation cannot feasibly be evaluated over patterns of full chip scale, since compute cost becomes prohibitive. Fortunately, the full chip problem can be reduced to one that scales near linearly with area by breaking the circuit level into parts, referred to herein as OPC frames, correction frames, simulation frames, or simply as frames, with these frames being processed quasi-independently using a large number of processors, for example 1024 or 2048 processors. These frames must be larger than the resolution of the optical system in order that the dimensional compensation provided to a given mask shape properly take into account the influence from all nearby mask shapes that are sufficiently close as to noticeably influence the image of the given shape. (Such influences are referred to as optical proximity effects.) In other contexts the resolution of an optical system usually refers to, e.g., the width of the central core of the lens point spread function, which is about 75 nanometers in modern lithographic systems. However, point spread functions have long tails that fall off slowly, and in the context of compensating a mask shape for optical proximity effects it is therefore necessary to consider the associated weak impact from relatively distant patterns. The distance range deemed relevant is referred to in computational lithography parlance as the ambit, optical ambit, or optical diameter [OD], and is typically 1 or 2 microns. The spatial domain tcc is calculated over this range. The size of the OPC frame should, as a minimum necessary condition, be set at least as large as the optical ambit in order to properly account for local content when providing dimensional compensation. However, sizing the frame at the limiting ambit value is inefficient, because it only provides sufficient buffering context to accurately compensate features within a very small area at the center of the frame. In practice the frame size is therefore set a few times larger than the ambit, e.g. the frame size might be set at 5 to 10 microns, with the outer region of the frame (within about one OD of the boundary) serving as a guard band. Results from within the guard band may simply be discarded, with the inner frame contents being retained, and with the frames being overlapped by e.g. twice the guard band width in order that each mask pattern falls within the retained region of one frame.
Since the frames overlap it is not possible to define the contents of one frame independently of its neighbors, and this enforced overlap allows the shared influence that the inner cores of adjacent frames have on the wafer image to be accounted for. Though it is generally impractical to simultaneously account for the entire network of frame interactions across a full chip, it is common practice to use multiple communicating processors that operate in parallel, so that the task of determining the dimensional compensation for a plurality of frames can be carried out simultaneously, with the number of interacting frames that are processed in this way being e.g. 4 or 16. To accomplish this task the integrated circuit layout may be divided into regions that are correspondingly larger than a frame, e.g., 4 or 16 times larger. However, in this approach the basic computational scale of the image calculation remains that of the frame rather than the larger region, meaning that the image must be calculated over (typically square) areas of, e.g., 5 microns or 10 microns in size (including guard bands). Such dimensions are still quite large compared to the core optical resolution of, e.g., 75 nanometers, and evaluation of the Hopkins equation over such areas becomes impractically slow, due to its nonlinear scaling.
To carry out MBOPC (a commonly used acronym for Model-Based OPC) or mask design it is therefore necessary to approximately match the images from the partially coherent system using simpler systems whose images can be calculated more quickly. In practice all such approaches in current use are variants of the so-called Optimal Coherent Systems (OCS) method, which approximately matches the partially coherent images from the lithographic system with a sum of images produced by predetermined coherent systems, to be described in more detail below. The method is very widely used, but goes by many different names besides OCS, such as the Optimal Coherent Approximation (OCA), or the Sum of Coherent Systems method (SOCS), or Coherent Decomposition.
In a coherent system the illumination is produced by a single source point, and so may take the form of a single plane wave once the coherent illumination is collimated onto the mask object. Such illumination by only a single independent beam causes all pairs of mask points in the doubled domain to fully interfere with one another, and thus the pairwise interference takes place with a common degree of coherence (namely 100%). Because of this common coherence the image contribution produced by all interactions of a given mask point with all the other mask points (meaning all points within the other dimension of the doubled Hopkins domain) can be summed separately, and then multiplied by itself to obtain the image intensity. As a result of this devolution to a single domain, the image amplitude produced by a coherent system (and thus by each coherent system in the OCS approximation of the partially coherent imaging system) is formed as a linear superposition of amplitude contributions from the various mask points (which is then squared to provide an image intensity), and mathematically this linear superposition can be represented as a linear convolution of a coherent kernel with the mask pattern. Linear convolution processes can be simulated very rapidly using Fast Fourier Transforms, meaning that calculation of the image contribution from an OCS coherent system can be carried out far more rapidly than direct calculation of the image produced by a partially coherent system. Even if the OCS set contains hundreds of coherent systems, it can be more efficient when dimensionally compensating mask shapes to approximately match the partially coherent image by the sum of hundreds of coherent images from the OCS set, instead of working directly with the partially coherent intensity. However, the efficiency gain will generally not be large enough to make dimensional compensation practical at full chip scale unless a set of coherent systems can be found which successfully match the partially coherent images with adequate accuracy using a somewhat smaller total number of coherent systems, e.g., if the OCS set achieves acceptable accuracy using only about 25 coherent systems.
The simplest approach for choosing coherent systems that match a partially coherent system is to subdivide the partially coherent source into small point-like elements. Point sources provide coherent illumination, and thus a separate coherent system can be defined for each grid point in a gridding of the complex source shape used by the partially coherent system. This simple decomposition into coherent systems was developed by Abbe to analyze microscope images, and is known as Abbe's method. The illumination for each coherent system essentially takes the form of a single plane wave that is incident from the direction of a particular source point. The projection lens in each coherent system collects the coherent light transmitted by the mask, and in the Abbe mode of coherent matching the lens apertures of these coherent systems are identical to the lens used by the partially coherent system being matched. However, rather than having the different coherent systems in the matching set use different illumination tilts along with a common lens aperture, one may equivalently use a common direction for the illuminating plane waves (such as illumination at normal incidence to the mask), while skewing the lens aperture to a different offset position for each of the different coherent systems. These two alternative sets of coherent systems behave equivalently because in Hopkins imaging the effect of tilting a plane wave that illuminates an object (e.g. the mask) is simply to introduce a matching directional skew or tilt in the plane waves that diffract from the mask, meaning that the set of collected waves can be changed in an equivalent way by either tilting the illumination, or by skewing the collection aperture to an offset position. Thus, in the Abbe approach each coherent system can be formed by shifting the lithographic lens aperture to a location that is offset to intersect the direction of some single point in the source, with the intensity contribution from each coherent system being weighted by the intensity of the associated source point (and with a common coherent plane wave illumination being used by all systems). Imposition of such a weighting factor is equivalent to introducing a uniform change in the transmission of the lens pupil of the coherent system.
In some cases this simple Abbe approach can provide an efficiency gain with the partially coherent sources used in modern lithography, since current sources are sparse in a relative sense, meaning that current lithographic sources only introduce significant illuminating intensity from a small fraction of the full range of directions from which the mask might in principle be illuminated (i.e. only small fraction of the full hemisphere of potentially incident directions actually contains illuminating waves). In the opposite extreme, i.e. when a mask is flood-illuminated with uniform intensity from a full hemisphere of directions, the illumination on the mask object becomes incoherent, and the pairwise contribution of object points to the image (as specified by the Hopkins equation) has magnitude zero unless the two points are coincident, i.e. are the same single mask point. The object effectively becomes self-luminous in this incoherent limit, and in this limit the doubled domain of the Hopkins reduces to a single domain. Incoherent images can therefore be calculated very rapidly using linear convolution of an intensity kernel with the self-luminous object. Imaging becomes incoherent when, for example, an object mask is flood-illuminated, or when an object is self-luminous, or when a self-luminous pattern is created by illuminating a fluorescent medium with a shaped pattern.
Lithographic sources are neither coherent nor incoherent, but they are usually considerably closer to the coherent limit than the incoherent limit, since the coherence function defined by modern sources shows appreciable content over distances that are distinctly larger than the projection lens resolution. Nonetheless, most lithographic systems remain quite far from even the coherent limit. In fact, a significant practical drawback to the simple Abbe coherent matching approach arises from the relatively large number of coherent systems that are needed to match typical partially coherent systems when Abbe decomposition is used. For example, a typical lithographic source shape can easily contain more than 100 source points that emit with significant intensity when an accurate gridding is used, and may contain 100's of additional points that emit with an intensity that is weak but non-zero, whose contributions should still be included to obtain accurate dimensional compensation. Use of such a large number of coherent systems forces an undesirably long compute time when determining appropriate dimensional compensations in patterns at full chip scale.
The inefficiency of the simple Abbe form of coherent decomposition arises from the very limited character of the tailoring that this method makes when defining each coherent system in the matching set, since the Abbe method attempts to provide a useful contribution to the match by simply shifting the position and uniform transmission of a lens pupil having fixed shape (i.e., in Abbe decomposition each coherent system aperture maintains the fixed shape of the circular pupil of the projection lens, except that it is shifted in position and given an adjusted transmission to match the contribution from a single emitting point of the partially coherent source).
It is known that a more efficient set of coherent systems can be obtained by employing coherent apertures that have complex general form, wherein the transmission of each pupil is made continuously varying in a complex pattern that yields the best possible match, rather than being merely a simple shifted disk. The transmission pattern of the lens aperture in a coherent system essentially acts as a filter on the diffracted mask spectrum, i.e. the aperture pattern applies a filtering to the mask spatial frequency content that the lens reconverges to the coherent image, and there is a known method for obtaining the coherent filter function which best matches the behavior of a partially coherent imaging system.
In particular, a method is known for determining the set of coherent system apertures which are optimally efficient, i.e., the set of aperture transmission functions which will be able to obtain a particular accuracy level using fewer coherent systems than any other set of apertures, when averaged over all possible patterns. Since coherent kernels are optimal when chosen in this way, they are referred to as Optimal Coherent Systems (OCS), and their use is also referred to as an Optimal Coherent Approximation (OCA), or as a Sum Of Coherent Systems (SOCS) approach.
When inverse Fourier transformed to the mask domain, these optimal coherent pupils become the kernels in linear convolutions of the mask patterns. It is known that these kernels may be explicitly determined as the eigenfunctions of the nonlinear (specifically, bilinear) kernel of the Hopkins equation, i.e., as eigenfunctions of the transmission cross coefficient (TCC, in the Fourier domain, or tcc, in the spatial domain, with lower case being used by convention in the latter acronym to denote a spatial domain quantity, and it is further known that the sum of an infinite number of squared convolutions of these eigenfunctions with the mask will reproduce the Hopkins equation result exactly. However, in practice OCS must use only a finite number of such squared convolutions, with each squared convolution providing the associated coherent image contribution, so that OCS approximately matches the partially coherent image using a finite sum of coherent images. The eigenfunction kernels used by OCS may be explicitly determined from the TCC by using standard algorithms and software packages for eigen decomposition. Some of these algorithms provide a complete eigen decomposition of the TCC when the TCC is gridded as a matrix, and since such a decomposition is akin to matrix diagonalization, the procedure is sometimes referred to as diagonalizing the TCC. At kernel counts that are practical for OPC (e.g., in the 10 to 100 range) the OCS coherent systems (each defined by a single kernel) provide a far more accurate match to the TCC than can coherent systems chosen by the Abbe method.
However, OCS accuracy still entails compromise in practice. For example, it should be noted that in a rigorous treatment the optical interaction range must be considered unbounded, though the interaction strength falls off rapidly to generally negligible levels once the core lens resolution and coherence length are exceeded. If the physical source contains an infinite number of points it would be necessary to use an infinite number of terms to exactly decompose the TCC into coherent system contributions, regardless of whether the Abbe or OCS method is used. However, at practical kernel counts the OCS method exhausts the TCC far more rapidly than the Abbe method (with the former in fact exhausting the TCC at the fastest rate possible for coherent systems), and the approximate match to the TCC that OCS provides is thus regarded as a valid decomposition of the exact TCC, even though it generally leaves unmatched a residual portion of the TCC whose impact on images of practical interest is often not entirely negligible.
Despite this imperfect accuracy, the OCS algorithm made MBOPC practical, and broadly speaking it has represented the state of the art in fast simulation of partially coherent projected images since about the mid-1990s. But even though OCS allows computational lithography shape adjustments to be determined with passable accuracy at speeds that are many orders of magnitude faster than is possible with direct evaluation of the Hopkins equation, MBOPC at full chip scale still requires very long compute times (of order one day) on very large computers, and so is quite expensive. Moreover, appreciable accuracy is often sacrificed in order to mitigate this high computational cost, and this increases the burden on empirical correction procedures that are used to fine-tune printed lithographic dimensions during production.
A further difficulty with the OCS algorithm is that the tradeoff between accuracy and speed becomes increasingly less favorable as required accuracy is tightened. Typical industry accuracy requirements have slowly increased as integrated circuit (IC) feature sizes push closer and closer to fundamental resolution limits, and this improvement requires a disproportionate increase in the number of OCS systems employed.
In summary, conventional practice to control the dimensions of IC patterns involves adjusting mask patterns in a process whose core is the so-called OCS method. The mask adjustment process (known as optical proximity correction or OPC) relies on OCS to assess candidate mask adjustments.
During use the OCS method simulates the wafer image of billions of mask features during each iteration of the adjustment. The OCS method constructs the wafer image as a sum of coherent images of the mask. Each coherent image in this approximate match to the partially coherent image is obtained as the squared convolution of the mask with a coherent kernel. The kernel may be considered as a function that is used as a component of an integration that is repeatedly applied, and each kernel in the OCS set is the inverse Fourier transform of the lens aperture of an Optimal Coherent System, which may be obtained as an eigenfunction of the Hopkins bilinear kernel. In general, a purely coherent image can be calculated as the squared convolution of a kernel with the object transmission, with the kernel being the inverse Fourier transform of the lens aperture (e.g. an Airy function in the simple case where the lens aperture is an open circle).
The current OPC practice requires a difficult tradeoff between runtime and accuracy when employing OCS. The OCS sum is only strictly accurate as an infinite series. However in current practice it may be considered reasonable to employ about 25 coherent systems to match the partially coherent lithographic system with an acceptable balance between runtime and accuracy, and therefore the OCS sum is typically terminated after only about 25 systems. An acceptable compromise can nonetheless involve too-large CD errors (typically ˜2 nm although larger errors can be experienced for some pitches) and too-slow runtimes (e.g., a day or more on a very large computer).
Clearly, improvements to the conventional OPC and OCS-based methods are needed.